They were killed a day after 14 members of the security forces died trying to retake a district in Libya's second city held by rival groups.
"Five soldiers were killed in an operation aimed at repelling terrorist groups in western Benghazi, as well as eight fighters belonging to these movements," said the agency which is close to the recognised authorities.
The oil-rich North African country has had rival administrations since the summer of 2014 when the recognised government fled Tripoli after a militia alliance including Islamists overran the capital.
"Orders were given to the soldiers to continue their advance and not to stop until they have successfully overcome the terrorist groups in Benghazi" in an operation dubbed "The Blood of Martyrs", the source said.
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For a year and a half, Benghazi has seen bloody fighting between armed groups including IS and Ansar al-Sharia, which is close to Al-Qaeda, and forces loyal to the government.
LANA, citing a medical source, also said seven people were killed in Ajdabiya (190 kilometres, 120 miles southwest of Benghazi) in two days of fighting between loyalist forces and rival groups.
The violence has allowed the jihadist Islamic State group to establish a foothold, and IS now controls the coastal city of Sirte and its surroundings.